Often the map q is understood, and Q itself is called the cokernel of f. In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f : X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.
In order for the definition to make sense the category in question must have zero morphisms.
Conversely an epimorphism is called normal (or conormal) if it is the cokernel of some morphism.
In the category of groups, the cokernel of a group homomorphism f : G → H is the quotient of H by the normal closure of the image of f. In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo the image of f: In a preadditive category, it makes sense to add and subtract morphisms.
The kernel may be expressed as the subspace (x, 0) ⊆ V: the value of x is the freedom in a solution.